Related papers: A splitter theorem on 3-connected matroids and gra…

We show that every 3-regular circle graph has at least two pairs of twin vertices; consequently no such graph is prime with respect to the split decomposition. We also deduce that up to isomorphism, K_4 and K_{3,3} are the only 3-connected,…

A vertex whose removal in a graph $G$ increases the number of components of $G$ is called a cut vertex. For all $n,c$, we determine the maximum number of connected induced subgraphs in a connected graph with order $n$ and $c$ cut vertices,…

It is shown that for a constant $t\in \mathbb{N}$, every simple topological graph on $n$ vertices has $O(n)$ edges if it has no two sets of $t$ edges such that every edge in one set is disjoint from all edges of the other set (i.e., the…

A hypergraph $H$ is hamiltonian-connected if for any distinct vertices $x$ and $y$, $H$ contains a hamiltonian Berge path from $x$ to $y$. We find for all $3\leq r<n$, exact lower bounds on minimum degree $\delta(n,r)$ of an $n$-vertex…

We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe…

An {\em odd hole} in a graph is an induced subgraph which is a cycle of odd length at least five. An {\em odd parachute} is a graph obtained from an odd hole $H$ by adding a new edge $uv$ such that $x$ is adjacent to $u$ but not to $v$ for…

A hole is an induced cycle of length at least 4. Let $\l\ge 2$ be a positive integer, let ${\cal G}_l$ denote the family of graphs which have girth $2\l+1$ and have no holes of odd length at least $2\l+3$, and let $G\in {\cal G}_{\l}$. For…

Halin proved that every minimally $k$-connected graph has a vertex of degree $k$. More generally, does every minimally vertically $k$-connected matroid have a $k$-element cocircuit? Results of Murty and Wong give an affirmative answer when…

By the theorem of Mantel $[5]$ it is known that a graph with $n$ vertices and $\lfloor \frac{n^{2}}{4} \rfloor+1$ edges must contain a triangle. A theorem of Erd\H{o}s gives a strengthening: there are not only one, but at least…

The isotropic matroid $M[IAS(G)]$ of a graph $G$ is a binary matroid, which is equivalent to the isotropic system introduced by Bouchet. In this paper we discuss four notions of connectivity related to isotropic matroids and isotropic…

The prism over a graph $G$ is the Cartesian product of $G$ with the complete graph $K_2$. A graph $G$ is hamiltonian if there exists a spanning cycle in $G$, and $G$ is prism-hamiltonian if the prism over $G$ is hamiltonian. In…

Let $G$ be a graph. A subset $D \subseteq V(G)$ is called a 1-isolating set of $G$ if $\Delta(G-N[D]) \leq 1$, that is, $G-N[D]$ consists of isolated edges and isolated vertices only. The $1$-isolation number of $G$, denoted by…

For all integers $k$ with $k\geq 2$, if $G$ is a balanced $k$-partite graph on $n\geq 3$ vertices with minimum degree at least \[…

The Union Closed Sets Conjecture is one of the most renowned problems in combinatorics. Its appeal lies in the simplicity of its statement contrasted with the potential complexity of its resolution. The conjecture posits that, in any union…

Given a graph $G$, denote by $h(G)$ the smallest size of a subset of $V(G)$ which intersects every maximum independent set of $G$. We prove that any graph $G$ without induced matching of size $t$ satisfies $h(G)\le \omega(G)^{3t-3+o(1)}$.…

A graph is prime (with respect to the split decomposition) if its vertex set does not admit a partition (A,B) (called a split) with |A|, |B| >= 2 such that the set of edges joining A and B induces a complete bipartite graph. We prove that…

There has been wide interest in understanding which properties of base graphs of matroids extend to base-cobase graphs of matroids. A significant result of Naddef and Pulleyblank (1984) shows that the $1$-skeleton of any $(0,1)$-polytope is…

Let $k \ge 3$ be an integer, $H_{k}(G)$ be the set of vertices of degree at least $2k$ in a graph $G$, and $L_{k}(G)$ be the set of vertices of degree at most $2k-2$ in $G$. In 1963, Dirac and Erd\H{o}s proved that $G$ contains $k$…

We say that a graph is intrinsically knotted or completely 3-linked if every embedding of the graph into the 3-sphere contains a nontrivial knot or a 3-component link any of whose 2-component sublink is nonsplittable. We show that a graph…

For a split graph $S$, the combinatorics of 2-switches on $S$ is faithfully encoded by the factor graph $\Phi(S)$, a multigraph whose induced cycles have length at most $4$. In this paper we address the following question: for which $n \in…